For two weighted random walks, \(X\)  and \(\bar{X}\)  on the same graph (but with different weights), one can relate the two relaxation times; by ([1]}, Lemma 3.29)
\( \tau _2 \le \bar{\tau }_2 \max _{v \in V}\frac{w(v)}{\bar{w}(v)}\max _{e \in E}\frac{\bar{w}(e)}{w(e)}.\) 
